This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present techniques. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the present techniques. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
Hydrocarbons are widely used for fuels and chemical feedstocks. Hydrocarbons may be found in subsurface rock formations that are generally termed reservoirs. Removing hydrocarbons from the reservoirs depends on numerous physical properties of the rock formations, such as the permeability of the rock containing the hydrocarbons, the ability of the hydrocarbons to flow through the rock formations, and the proportion of hydrocarbons present, among others. Structural changes in the geological materials affected by reservoir production can have a significant effect on the performance of a wellbore. For example, reservoir compaction can affect pore pressure and rock porosity, resulting in a change in the flow rate of hydrocarbons moving through the rock.
Geomechanical modeling of geological structures provides techniques for analyzing the possible effects of reservoir production on the immediate and surrounding geological structures. Thus, geomechanical modeling can play an important role in analyzing various aspects of hydrocarbon reservoirs. Geomechanical modeling can also be used in forward modeling and restoration modeling of the evolution of geological structures at geologic time scales (i.e. millions of years). Geomechanical modeling can also be used to predict wellbore performance, reservoir compaction, pore pressure, and the like.
The geomechanical model can include the geomechanical properties of subsurface materials, referred herein as “geomaterials.” The geomaterials are rarely homogenous, highly variable in their properties, and also contain discontinuities such as fractures. The sources of heterogeneity are environment of deposit (EOD), stratification, diagenetic history, tectonic loading histories, and the like. The scale of heterogeneity is often very small, for example, on the order of a centimeter. Additionally, the mechanical properties of the rocks are scale-dependent. See e.g, Barton, et al., “Engineering Classification of Rock Masses for the Design of Tunnel Support,” 6:189-236 ROCK MECHANICS (1974); Hoek & Brown, Underground Excavations in Rock, INSTITUTION OF MINING & METALLURGY, (1980); Suárez-Rivera, et al., “Continuous Rock Strength Measurements on Core and Neural Network Modeling Result in Significant Improvements in Log-Based Rock Strength Predictions Used to Optimize Completion Design and Improve Prediction of Sanding Potential and Wellbore Stability,” SPE 84558, SOCIETY OF PETROLEUM ENGINEERS (2003).
Such fine-scale variation of geomechanical properties may be captured in the geological models using rock property algorithms which are developed by integrating data from seismic, well logs, EOD models, lithology, porosity, core tests, and petrophysical characterization. See, U.S. Provisional Patent Application No. 61/226,999, by Crawford, et al. The geological model can also include discontinuities such as fractures. The orientation, spacing, and interfacial properties of discontinuities contribute to the scale dependency of the rock properties.
The mechanical properties of geomaterials may also depend on the drainage conditions of the geomaterials. In an easily drained rock, fluid is able to travel out of the rocks very quickly, which results in little or no excess pore fluid pressure. In a poorly drained rock, fluid is not able to drain out of rocks quickly, resulting in the development of a finite amount of pore pressure within the rock.
To characterize the fine-scale variation of mechanical properties, it would be beneficial to build the geomechanical model at scales close to the geological models. However, simulating such a large scale geomechanical model would involve an enormous amount of computational resources. Even with present-day computational power, geomechanical analyses, especially in three dimensions, cannot be performed at a fine scale. Thus, typical geomechanical analyses are performed at coarse scale. For example, a typical geomechanical model cell-size is about tens to hundreds of meters in the horizontal directions and few meters in the vertical direction. A single geomechanical model cell can be used to represent about 100 to 1000 corresponding geological model cells with varying geomechanical properties. In the numerical methods used for geomechanical analyses, each coarse geomechanical cell is characterized by a single set of mechanical properties. The process of determining a single set of properties for coarse-scale geomechanical cells using the corresponding properties of fine-scale cells in the geological model is often referred to as homogenization or upscaling.
Upscaling in various forms has been used in various branches of engineering. An estimation of upscaled elastic modulus of mixtures is one of the classical problems in the micromechanics. See e.g., Voigt, “Uber die Beziehung zwischen den beiden Elastizitatkonstanten isotroper Korper,” 38:573-587 ANNALS OF PHYSICS (1889); Reuss, “Calculation of the Flow Limits of Mixed Crystals on the Basis of Plasticity of Mono-crystals,” 9:49-58 ZEITSCHRIFT FUER ANGEWANDTE MATHEMATIK UND MECHANIK (1929).
Upper and lower bounds of elastic modulus of mixtures has been derived on the basis of the variational principles and compared to numerical answers with experimental results. See e.g., Hashin, “The Elastic Moduli of Heterogeneous Materials,” 10:143-150 JOURNAL OF APPLIED MECHANICS (1962); Hashin & Shtrikman, “On Some Variational Principles in Anisotropic and Nonhomogeneous Elasticity,” 10: 335-342 JOURNAL OF MECHANICS AND PHYSICS OF SOLIDS (1962). These methods are not relevant for reservoir rocks as they may not be idealized as elastic mixtures.
Upscaling methods have been developed for perfectly layered rocks. See Backus, “Long-wave Elastic Anisotropy Produced by Horizontal Layering,” 67: 4427-4440 JOURNAL OF GEOPHYSICAL RESEARCH (1962). Additionally, equations have been developed for upscaled properties of a homogeneous transversely isotropic medium of horizontally layered rock. See Salamon, “Elastic Moduli of a Stratified Rock Mass,” 5:591-527 JOURNAL OF ROCK MECHANICS AND MINING SCIENCES (1968). Further, these methods have been extended to orthorhombic layers. See Gerrard, “Equivalent Elastic Moduli of a Rock Mass Consisting of Orthorhombic Layers,” 19:9-14 JOURNAL OF ROCK MECHANICS AND MINING SCIENCES AND GEOMECHANICS (1982). These methods have been further improved for the case of imperfectly layered rock types. See Rijpsma & Zijl, “Upscaling of Hooke's Law for Imperfectly Layered Rocks,” 30:943-969 MATHEMATICAL GEOLOGY (1998). The above publications assume that the rock properties are homogeneous within each layer. Therefore, the methods described in those publications are not suitable for upscaling geomechanical properties of reservoir rocks that exhibit a significant degree of heterogeneity within individual layers. These methods are not suitable for incorporating discontinuities such as fractures and not suitable for study of impact of pore fluid drainage on mechanical properties.
Another Upscaling approach has been developed that uses a node-based finite element method with displacement as the primary variable. Zijl et al., “Numerical Homogenization of the Rigidity Tensor in Hooke's Law Using the Node-Based Finite Element Method,” 34:291-322 MATHEMATICAL GEOLOGY (2002). An upscaled elastic rigidity tensor is computed from volume averaged stress and strain such that the equilibrium, elastic law, and energy preserve their scale form on the coarse scale. The method gives an upper bound of the elastic rigidity tensor for a given number of elements. The main limitation of this method is that the discontinuities which make rock properties scale dependent are not considered. A similar method has been developed to for computing the elastic rigidity tensor. Chalon, et al., “Upscaling of Elastic Properties for Large Scale Geomechanical Simulations,” 28:1105-1119 INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS (2004).
Further, U.K. Patent No. GB2413200 by Mainguy, et al. and U.S. Patent Publication 2005/02346909 by Mainguy, et al. disclose methods of constructing a geomechanical model using methods to upscale elastic properties of rocks. Those methods address upscaling of only elastic properties and do not address upscaling of plastic properties. Furthermore, those methods do not consider the presence of discontinuities, such as fractures, which make rock properties scale dependent. The methods described also do not address deriving rock properties that are influenced by drainage of fluids.
None of the techniques described above provide a generalized method that is suitable for upscaling both elastic and plastic properties of subsurface rocks. Further, none of the techniques described above account for discontinuities or pore fluid drainage.